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This paper is concerned with the distributed optimal control of a
time-discrete Cahn--Hilliard/Navier--Stokes system with variable
densities.
It focuses on the double-obstacle potential which yields an optimal
control problem for a family of coupled systems in each time instance of a
variational inequality of fourth order and the Navier--Stokes equation.
By proposing a suitable time-discretization, energy estimates are proved
and the existence of solutions to the primal system and of optimal
controls is established for the original problem as well as for a family
of regularized problems. The latter correspond to Moreau--Yosida type
approximations of the double-obstacle potential. The consistency of these
approximations is shown and first order optimality conditions for the
regularized problems are derived. Through a limit process, a stationarity
system for the original problem is established which is related to a
function space version of C-stationarity.
A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static
elasto-plasticity with kinematic or isotropic hardening is considered. The associated path is induced
by a combined Moreau-Yosida / Tichonov regularization of the dual problem.
The sequence of solutions to the regularized problems is shown to converge strongly to the optimal
displacement-stress-strain triple of the original elasto-plastic contact problem in the space-continuous setting.
This property relies on the density of the intersection of certain convex sets which is shown as well.
It is also argued that the mappings associated with the resulting problems are Newton- or slantly differentiable.
Consequently, each regularized subsystem can be solved mesh-independently at a local
superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and
a numerical validation of the theoretical results is given.
A mechanical equilibrium problem for a material consisting of two components with dierent densities is considered. Due to the heterogeneous material densities, the
outer shape of the underlying workpiece can be changed by shifting the interface between
the subdomains. In this paper, the problem is modeled as a shape design problem for optimally compensating unwanted workpiece changes. The associated control variable is
the interface. Regularity results for transmission problems are employed for a rigorous
derivation of suitable first-order optimality conditions based on the speed method. The paper concludes with several numerical results based on a spline approximation of the
interface.